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Theorem neldif 3391
 Description: Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)
Assertion
Ref Expression
neldif ((A B ¬ A (B C)) → A C)

Proof of Theorem neldif
StepHypRef Expression
1 eldif 3221 . . . 4 (A (B C) ↔ (A B ¬ A C))
21simplbi2 608 . . 3 (A B → (¬ A CA (B C)))
32con1d 116 . 2 (A B → (¬ A (B C) → A C))
43imp 418 1 ((A B ¬ A (B C)) → A C)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∈ wcel 1710   ∖ cdif 3206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215 This theorem is referenced by: (None)
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